# Black hole entropy as a Noether Charge

1. Mar 28, 2007

### haushofer

Hi folks, I have a question about a paper by Thomas Mohaupt, called
"Black hole entropy, special geometry and strings". It's available here:

http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3Ahep-th%2F0007195 [Broken]

My question concerns part 2.2.5 , page 18. I find it quite difficult to get a nice feeling for doing calculations like the ones in that part. Here the author defines entropy as a surface charge ( a method due to Wald ). He assumes that the Lagrangian depends on the Riemanntensor, the energy-momentum (E-M) tensor and the derivative of the E-M tensor ( the covariant, I presume? ). He then considers a variation of the E-M tensor and the metric, which I do understand ( they're defined via a general coordinate transformation, so you end up with a Lie-derivative ) So up to that it's okay.

First, I want to do the variation of the Lagrangian with respect to the Riemanntensor, and here I get some troubles. How do I write down such a variation ? It looks like

$$\delta S = \frac{\partial L }{\partial R_{\mu\nu\rho\sigma} } \delta R_{\mu\nu\rho\sigma}$$

Here L is the Lagrangian density.
If I want to rewrite this in terms of the variation with respect to the metric, can I use the usual chain rules for differentiation? Or should I do the differentiation explicitly with respect to the metric, and rewrite this in terms of the Riemanntensor?

Then they define a current J. Via Noethers theorem I know that one can define such a current as

$$J^{\mu} = \frac{\partial L}{\partial \phi ,\mu} \delta \phi$$

where we sum over all fields phi. So my guess would be that the current which is written down there in section 2.2.5 is acquired via

$$J^{\mu} = \frac{\partial L}{\partial R_{\nu\lambda\rho\sigma},\mu} \delta R_{\nu\lambda\rho\sigma} + \frac{\partial L}{\partial \psi_{\nu\lambda},\mu} \delta \psi_{\nu\lambda}$$

Is this going into the right direction? In the variation of the metric and the E-M tensor they use a test function epsilon which is a function of the coordinates, but I don't see it in the current back. That's strange, because you can't divide the function out of the current ( we get derivatices of the test function ). Or can we put those terms to 0? Also I have some doubts about when to use partial derivatives in such calculations, and when to use covariant derivatives. Should I just replace al partial derivatives in such calculations by covariant derivatives?

And does any-one know a good text(book) in which all this is nicely explained? I've been searching on the internet, but without good results. A lot of questions, I hope some-one can help me. Many thanks in forward,

Haushofer.

Last edited by a moderator: May 2, 2017
2. Apr 2, 2007

Nobody ? :(